Schauder Basis

A Schauder basis for an infinite-dimensional Banach space over or is a sequence such that every has a unique expansion , with scalars , converging in the norm topology. Explicitly,

In a finite-dimensional Banach space, the corresponding finite notion is an ordinary vector basis. A Schauder expansion permits infinitely many nonzero coefficients, while each expansion in an algebraic vector basis is a finite linear combination (Heil 1998, pp. 39-40).

The coefficient maps are continuous linear functionals (Heil 1998, p. 43).

For example, the trigonometrical system is a basis in each L^p-space on for .